Beth number

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In mathematics, the infinite cardinal numbers are represented by the Hebrew letter [\aleph] (aleph) indexed with a subscript that runs over the ordinal numbers (see aleph number). The second Hebrew letter [\beth] (beth) is also used. To define the beth numbers, start by letting

[\beth_0=\aleph_0]

be the cardinality of any countably infinite set; for concreteness, take the set [\mathbb] of natural numbers to be a typical case. Denote by P(A) the power set of A, i.e., the set of all subsets of A. Then define

[\beth_=2^,]

which is the cardinality of the power set of A if [\beth_\kappa] is the cardinality of A.

Then

[\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots]

are respectively the cardinalities of

[\mathbb,\ P(\mathbb),\ P(P(\mathbb)),\ P(P(P(\mathbb))),\ \dots.]

One can also show that the von Neumann universes [V_] have cardinality [\beth_\alpha].

Each set in this sequence has cardinality strictly greater than the one preceding it, because of Cantor's theorem. Note that the 1st beth number [\beth_1] is equal to c (or [\mathfrak c]), the cardinality of the continuum, and the 2nd beth number [\beth_2] is the cardinality of the power set of the continuum.

For infinite limit ordinals κ, we define

[\beth_\kappa=\sup\].

If we assume the axiom of choice, then infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable, and so, since by definition no infinite cardinalities are between [\aleph_0] and [\aleph_1], the celebrated continuum hypothesis can be stated in this notation by saying

[\beth_1=\aleph_1.]

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers.

The more general symbol [\beth_\kappa(\alpha)], for ordinals κ and cardinals α, is occasionally used. It is defined by

[\beth_0(\alpha)=\alpha]

[\beth_(\alpha)=2^,]

[\beth_\kappa(\alpha)=\sup\] if κ is a limit ordinal.

So [\beth_\kappa=\beth_\kappa(\aleph_0).]

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